{
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   "source": [
    "# Quantization of Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Characteristic of a Linear Uniform Quantizer\n",
    "\n",
    "The characteristics of a quantizer depend on the mapping functions $f(\\cdot)$, $g(\\cdot)$ and the rounding operation $\\lfloor \\cdot \\rfloor$ introduced in the [previous section](introduction.ipynb). A linear quantizer bases on linear mapping functions $f(\\cdot)$ and $g(\\cdot)$. A uniform quantizer splits the mapped input signal into quantization steps of equal size. Quantizers can be described by their nonlinear in-/output characteristic $x_Q[k] = \\mathcal{Q} \\{ x[k] \\}$, where $\\mathcal{Q} \\{ \\cdot \\}$ denotes the quantization process. For linear uniform quantization it is common to differentiate between two characteristic curves, the so called mid-tread and mid-rise. Both are introduced in the following."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mid-Tread Characteristic Curve\n",
    "\n",
    "The in-/output relation of the mid-tread quantizer is given as\n",
    "\n",
    "\\begin{equation}\n",
    "x_Q[k] = Q \\cdot \\underbrace{\\left\\lfloor \\frac{x[k]}{Q} + \\frac{1}{2} \\right\\rfloor}_{index}\n",
    "\\end{equation}\n",
    "\n",
    "where $Q$ denotes the constant quantization step size and $\\lfloor \\cdot \\rfloor$ the [floor function](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions) which maps a real number to the largest integer not greater than its argument. Without restricting $x[k]$ in amplitude, the resulting quantization indexes are [countable infinite](https://en.wikipedia.org/wiki/Countable_set). For a finite number of quantization indexes, the input signal has to be restricted to a minimal/maximal amplitude $x_\\text{min} < x[k] < x_\\text{max}$ before quantization. The resulting quantization characteristic of a linear uniform mid-tread quantizer is shown below\n",
    "\n",
    "![Characteristic of a linear uniform mid-tread quantizer](mid_tread_characteristic.png)\n",
    "\n",
    "The term mid-tread is due to the fact that small values $|x[k]| < \\frac{Q}{2}$ are mapped to zero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Mid-tread quantization of a sine signal\n",
    "\n",
    "The quantization of one period of a sine signal $x[k] = A \\cdot \\sin[\\Omega_0\\,k]$ by a mid-tread quantizer is simulated. $A$ denotes the amplitude of the signal, $x_\\text{min} = -1$ and $x_\\text{max} = 1$ are the smallest and largest output values of the quantizer, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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       "     </g>\n",
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       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pc2f0c7c54a\">\n",
       "   <rect height=\"332.64\" width=\"558\" x=\"38.482813\" y=\"7.2\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 960x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "A = 1.2  # amplitude of signal\n",
    "Q = 1/10  # quantization stepsize\n",
    "N = 2000  # number of samples\n",
    "\n",
    "\n",
    "def uniform_midtread_quantizer(x, Q):\n",
    "    # limiter\n",
    "    x = np.copy(x)\n",
    "    idx = np.where(np.abs(x) >= 1)\n",
    "    x[idx] = np.sign(x[idx])\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * np.floor(x/Q + 1/2)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "def plot_signals(x, xQ):\n",
    "    e = xQ - x\n",
    "    plt.figure(figsize=(10, 6))\n",
    "    plt.plot(x, label=r'signal $x[k]$')\n",
    "    plt.plot(xQ, label=r'quantized signal $x_Q[k]$')\n",
    "    plt.plot(e, label=r'quantization error $e[k]$')\n",
    "    plt.xlabel(r'$k$')\n",
    "    plt.axis([0, N, -1.1*A, 1.1*A])\n",
    "    plt.legend()\n",
    "    plt.grid()\n",
    "\n",
    "\n",
    "# generate signal\n",
    "x = A * np.sin(2*np.pi/N * np.arange(N))\n",
    "# quantize signal\n",
    "xQ = uniform_midtread_quantizer(x, Q)\n",
    "# plot signals\n",
    "plot_signals(x, xQ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Change the quantization stepsize `Q` and the amplitude `A` of the signal. Which effect does this have on the quantization error?\n",
    "\n",
    "Solution: The smaller the quantization step size, the smaller the quantization error is for $|x[k]| < 1$. Note, the quantization error is not bounded for $|x[k]| > 1$ due to the clipping of the signal $x[k]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mid-Rise Characteristic Curve\n",
    "\n",
    "The in-/output relation of the mid-rise quantizer is given as\n",
    "\n",
    "\\begin{equation}\n",
    "x_Q[k] = Q \\cdot \\Big( \\underbrace{\\left\\lfloor\\frac{ x[k] }{Q}\\right\\rfloor}_{index}  + \\frac{1}{2} \\Big)\n",
    "\\end{equation}\n",
    "\n",
    "where $\\lfloor \\cdot \\rfloor$ denotes the floor function. The quantization characteristic of a linear uniform mid-rise quantizer is illustrated below\n",
    "\n",
    "![Characteristic of a linear uniform mid-rise quantizer](mid_rise_characteristic.png)\n",
    "\n",
    "The term mid-rise copes for the fact that $x[k] = 0$ is not mapped to zero. Small positive/negative values around zero are mapped to $\\pm \\frac{Q}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Mid-rise quantization of a sine signal\n",
    "\n",
    "The previous example is now reevaluated using the mid-rise characteristic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 960x576 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "A = 1.2  # amplitude of signal\n",
    "Q = 1/10  # quantization stepsize\n",
    "N = 2000  # number of samples\n",
    "\n",
    "\n",
    "def uniform_midrise_quantizer(x, Q):\n",
    "    # limiter\n",
    "    x = np.copy(x)\n",
    "    idx = np.where(np.abs(x) >= 1)\n",
    "    x[idx] = np.sign(x[idx])\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * (np.floor(x/Q) + .5)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "# generate signal\n",
    "x = A * np.sin(2*np.pi/N * np.arange(N))\n",
    "# quantize signal\n",
    "xQ = uniform_midrise_quantizer(x, Q)\n",
    "# plot signals\n",
    "plot_signals(x, xQ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* What are the differences between the mid-tread and the mid-rise characteristic curves for the given example?\n",
    "\n",
    "Solution: The mid-tread and the mid-rise quantization of the sine signal differ for signal values smaller than half of the quantization interval. Mid-tread has a representation of $x[k] = 0$ while this is not the case for the mid-rise quantization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
 ],
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